Background flow hidden in a bound for Nusselt number

The well-known background flow method for finding bounds for time-averaged characteristics of dynamical systems, proposed by Doering and Constantin (1994, 1995) is a special case of the auxiliary functional method of Chernyshenko et al. (2014). Chernyshenko (2022) proved that bounds obtained by the direct method described by Seis (2015) can be obtained also by the auxiliary functional method and, therefore, by the background flow method when the auxiliary functional is quadratic. This brief note outlines the technique by which the background flow and more generally the auxiliary functional can be obtained when a proof of a bound for infinite time average by the direct method is known, by applying this technique to the case of the bound on the Nusselt number for infinite-Prandtl-number Rayleigh–Bénard convection obtained by Otto and Seis (2011).